Dynamic Kernels for Hitting Sets and Set Packing

Abstract

AbstractComputing small kernels for the hitting set problem is a well-studied computational problem where we are given a hypergraph with n vertices and m hyperedges, each of size d for some small constant d, and a parameter k. The task is to compute a new hypergraph, called a kernel, whose size is polynomial with respect to the parameter k and which has a size-k hitting set if, and only if, the original hypergraph has one. State-of-the-art algorithms compute kernels of size $$k^d$$ k d (which is a polynomial as d is a constant), and they do so in time $$m\cdot 2^d {\text {poly}}(d)$$ m · 2 d poly ( d ) for a small polynomial $${\text {poly}}(d)$$ poly ( d ) (which is linear in the hypergraph size for d fixed). We generalize this task to the dynamic setting where hyperedges may continuously be added or deleted and one constantly has to keep track of a size-$$k^d$$ k d kernel. This paper presents a deterministic solution with worst-case time $$3^d {\text {poly}}(d)$$ 3 d poly ( d ) for updating the kernel upon inserts and time $$5^d {\text {poly}}(d)$$ 5 d poly ( d ) for updates upon deletions. These bounds nearly match the time $$2^d {\text {poly}}(d)$$ 2 d poly ( d ) needed by the best static algorithm per hyperedge. Let us stress that for constant d our algorithm maintains a hitting set kernel with constant, deterministic, worst-case update time that is independent of n, m, and the parameter k. As a consequence, we also get a deterministic dynamic algorithm for keeping track of size-k hitting sets in d-hypergraphs with update times O(1) and query times $$O(c^k)$$ O ( c k ) where $$c = d - 1 + O(1/d)$$ c = d - 1 + O ( 1 / d ) equals the best base known for the static setting.